In this section, we give some basic concepts and facts that will be needed later.
Let \(\hbox{\rm Cay}(G,S)\)  be a Cayley digraph of a group \(G\)  with respect to \(S\) . Recall that \(\hbox{\rm Aut}(G,S)=\lbrace  \alpha \in \hbox{\rm Aut}(G)~|~S^{\alpha }=S\rbrace \) . Then \(\hbox{\rm Aut}(G,S)\)  is a subgroup of \(\hbox{\rm Aut}(\hbox{\rm Cay}(G,S))\) , and in fact, it is a subgroup of the stabilizer \((\hbox{\rm Aut}(\hbox{\rm Cay}(G,S)))_1\)  of 1 in \(\hbox{\rm Aut}(\hbox{\rm Cay}(G,S))\) . Godsil [1]} proved that the normalizer of \(R(G)\)  in \(\hbox{\rm Aut}(\hbox{\rm Cay}(G,S))\)  is a semiproduct of \(R(G)\)  by \(\hbox{\rm Aut}(G,S)\) , and then one may prove the following proposition easily (see [2]}).
